A conic bundle degenerating on the Kummer surface

Michele Bolognesi

Home/Archive/arXiv

arXiv · math/0702525

A conic bundle degenerating on the Kummer surface

Michele Bolognesi

View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $C$ be a genus 2 curve and $\su$ the moduli space of semi-stable rank 2 vector bundles on $C$ with trivial determinant. In \cite{bol:wed} we described the parameter space of non stable extension classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf $ω$ of $C$ with $ω_C^{-1}$. In this paper we study the classifying rational map $ϕ: \pr Ext^1(ω,ω^{-1})\cong \pr^4 \dashrightarrow \su\cong \pr^3$ that sends an extension class on the corresponding rank two vector bundle. Moreover we prove that, if we blow up $\pr^4$ along a certain cubic surface $S$ and $\su$ at the point $p$ corresponding to the bundle $\OO \oplus \OO$, then the induced morphism $\tildeϕ: Bl_S \ra Bl_p\su$ defines a conic bundle that degenerates on the blow up (at $p$) of the Kummer surface naturally contained in $\su$. Furthermore we construct the $\pr^2$-bundle that contains the conic bundle and we discuss the stability and deformations of one of its components.

Explore related subjects

math.AG

Keep this discovery

BibTeX RIS

Michele Bolognesi. 2007-02-18. A conic bundle degenerating on the Kummer surface. https://arxiv.org/abs/math/0702525

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Perverse Euler Characteristics of Hermitian Locally Symmetric Spaces

We prove that finite-volume locally Hermitian symmetric spaces of noncompact type have nonnegative perverse Euler characteristics. To show this, we obtain a nefness result for the logarithmic cotangent bundle of a smooth toroidal compactification. Combining this with a positivity criterion for Euler characteristics of perverse sheaves, we deduce the nonnegativity result. We further prove that the inequality is strict for perverse sheaves with full support. As applications, we get nonnegativity results for perverse Euler characteristics on various moduli spaces.

math.AG↗

arXiv2026

Coupled Pklt Tuples and Varieties of Pklt Type

We introduce asymptotic multiplier ideal sheaves and log canonical thresholds associated with tuples of pseudoeffective divisors on a projective klt pair. We prove that the threshold of a coupled potentially klt tuple is computed by a quasi-monomial valuation. For varieties of potentially klt type, we prove that every big divisor admits a birational Zariski decomposition with semiample positive part. We also prove finite generation of multisection rings of big divisors and give a criterion for a variety of potentially klt type to be a Mori dream space.

math.AG↗

arXiv2026

Graded Betti numbers of general curves of large degree

Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $\kappa_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $\kappa_{p,q}(C, L)$ when the Brill--Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes \omega_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $\deg L \geq 4g-3$ or when $\deg L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij--S\"{o}derberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.

math.AG↗